An inflated dynamic Laplacian to track the emergence and disappearance of semi-material coherent sets

Lagrangian methods continue to stand at the forefront of the analysis of time-dependent dynamical systems. Most Lagrangian methods have criteria that must be fulfilled by trajectories as they are followed throughout a given finite flow duration. This key strength of Lagrangian methods can also be a limitation in more complex evolving environments. It places a high importance on selecting a time window that produces useful results, and these results may vary significantly with changes in the flow duration. We show how to overcome this drawback in the tracking of coherent flow features. Finite-time coherent sets (FTCS) are material objects that strongly resist mixing in complicated nonlinear flows. Like other materially coherent objects, by definition they must retain their coherence properties throughout the specified flow duration. Recent work [Froyland and Koltai, CPAM, 2023] introduced the notion of semi-material FTCS, whereby a balance is struck between the material nature and the coherence properties of FTCS. This balance provides the flexibility for FTCS to come and go, merge and separate, or undergo other changes as the governing unsteady flow experiences dramatic shifts. The purpose of this work is to illustrate the utility of the inflated dynamic Laplacian introduced in [Froyland and Koltai, CPAM, 2023] in a range of dynamical systems that are challenging to analyse by standard Lagrangian means, and to provide an efficient meshfree numerical approach for the discretisation of the inflated dynamic Laplacian.